Page 264 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

P1: GLM/GLT P2: GLM Final
Encyclopedia of Physical Science and Technology En006G-249 June 27, 2001 14:7

56 Fluid Dynamics (Chemical Engineering)

inside and outside the pipe; A s is the area of the heated r dp
pipe surface; and T m is some sort of mean or average τ rz = 2 − dz = ξτ w , (70)
temperature difference between the fluid and the pipe wall.
where −dp/dz is the axial pressure gradient, ξ =r/R is
Depending on the definition of T m , the definitions of
a normalized radial position variable, and τ w is the wall
the local heat transfer coefficients vary and so does the
shear stress given by
definition of U m . This equation is not discussed further
here, as its full discussion properly belongs in a separate D p
article devoted to the subject of heat transfer. τ w = 4 − L . (71)
Equation (70) is clearly independent of any constitutive
IV. LAMINAR FLOW
relation and applies universally to all fluids in a pipe of
this geometry.
In laminar flow the velocity distribution, and hence the
frictional energy loss, is governed entirely by the rheolog-
ical constitutive relation of the fluid. In some cases it is 2. Concentric Annulus
possible to derive theoretical expressions for the friction
Suppose a solid core were placed along the centerline of
factor. Where this is possible, a three-step procedure must
the pipe described in the preceding section so as to be
be followed.
coaxial and concentric with the pipe. Equation (69) is still
valid as the solution of Eqs. (17)–(19). Now, however, the
1. Solve the equations of motion for the stress
point r = 0 is not included in the domain of the solution,
distribution.
so that C is no longer zero. Somewhere between the two
2. Couple the stress distribution with the constitutive
boundaries r = R i and r = R the shear stress will vanish.
relation to produce a differential equation for the
If this point is called ξ = λ, then Eq. (69) becomes
velocity field. Solve this equation for the velocity
distribution. τ rz = τ R (ξ − λ /ξ), (72)
2
3. Integrate the velocity distribution over the cross
where τ R is the shear stress at the outer pipe wall given by
section of the duct to obtain an expression for the
average velocity v . Rearrange this expression into a R dp
dimensionless form involving a friction factor. τ R = − (73)
2 dz
A. Shear Stress Distributions and ξ =r/R as before. Note two things: (1) Eq. (72) is
now nonlinear in ξ, and (2) we still do not know the value
In some special cases it is possible to solve the equations
of C. All that has been done is to shift the unknown value
of motion [Eq. (11)] entirely independently of any knowl-
of C to the still unknown value of λ. We do, however,
edge of the constitutive relation and to obtain a universal
know the physical significance of λ. It is the location of
shear stress distribution that applies to all fluids. In other the zero-stress surface. Unfortunately, we cannot discover
cases it is not possible to do this because the evaluation
the value of λ until we introduce some specific constitutive
of certain integration constants requires knowledge of the
relation,integratetheresultingdifferentialequationforthe
specificconstitutiverelation.Becauseofspacelimitations,
velocitydistribution(thusintroducingyetanotherconstant
we illustrate only one case of each type here.
ofintegration),andtheninvoketheno-sliporzero-velocity
boundary conditions at both solid boundaries to determine
1. Pipes
the values of the new integration constant and λ. The value
Equation (11) for the cylindrical geometry appropriate to of λ so determined will be different for each different
the circular cross-section pipes so commonly used in prac- constitutive relation employed.
tical situations is expressed by Eqs. (17)–(19). For steady,
fully developed, incompressible flow, the solution of these B. Velocity Distributions
equations is 1. Newtonian
r dp C
τ rz = − + , (69) When the Newtonian constitutive relation is coupled with
2 dz r Eq. (70) and appropriate integrations are performed, we
where C is a constant of integration. Considerations of obtain
boundedness at the pipe centerline, r = 0, require that 2
u = v z / v = 2(1 − ξ ) (74)
C = 0. Thus, Eq. (69) reduces to the familiar linear stress
distribution, v = Dτ w /8µ (75)